Hardy's Uncertainty Principle Research Articles

A metaplectic perspective of uncertainty principles in the linear canonical transform domain

We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schrödinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms.

An uncertainty principle for the windowed Bochner–Fourier transform with the complex-valued window function

Abstract The generalization of Hardy uncertainty principle for the windowed Bochner–Fourier transform on the Heisenberg group is proved. We consider a Bochner measurable function Ψ : G → X {\Psi:G\to X} , where X is a completely separable Hilbert space X and Let G be a completely separable, unimodular, connected nilpotent Lie group. We establish that if ϕ ∈ C C ⁢ ( G ) {\phi\in C_{C}(G)} is a non-trivial window function and Ψ ∈ L 2 ⁢ ( G ) {\Psi\in L^{2}(G)} satisfies ∥ V ϕ ⁢ ( Ψ ) ⁢ ( g , χ ) ∥ HB ≤ c 2 ⁢ ( g ) ⁢ exp ⁡ ( - β ⁢ ∥ χ ∥ 2 ) \|V_{\phi}(\Psi)(g,\chi)\|_{\rm HB}\leq c_{2}(g)\exp(-\beta\|\chi\|^{2}) , β > 0 {\beta>0} , then Ψ = 0 {\Psi=0} almost everywhere.

On the fractional space-time Fourier transforms

ABSTRACT The notion of a fractional space-time Fourier transform (FSFT) is outlined in this paper, and the properties of invertibility, linearity, Plancherel and others are derived. By establishing the relationship between the FSFT and space-time Fourier transform, a directional uncertainty principle (UP) and its specialization to coordinates are proved. Moreover, the Hardy UP of the FSFT is also obtained. The fractional Mustard convolution for space-time valued signals is introduced and written in terms of the standard convolution. Finally, the FSFT is employed in solving a partial differential equation in space-time analysis.

Notes on Hardy's uncertainty principle for the Wigner distribution and Schrödinger evolutions

For Schrödinger equations with real quadratic Hamiltonians, it is known that the Wigner distribution of the solution at a given time equals, up to a linear coordinate transformation, the Wigner distribution of the initial condition. Based on Hardy's uncertainty principle for the joint time-frequency representation, we present a general uniqueness result for such Schrödinger equations, where the solution cannot have strong decay at two distinct times. This approach gives new proofs to known, sharp Hardy type estimates for the free Schrödinger equation, the harmonic oscillator and uniform magnetic potentials, as well as new uniqueness results.

The uncertainty principle for the octonion Fourier transform

The octonion Fourier transform (OFT) is a hypercomplex Fourier transform that generalizes the quaternion Fourier transform. However, in octonion algebra, there are two major obstacles that are presented in the loss of associativity and commutativity. Researchers have been trying to extend the results of the Euclidean Fourier transform to quaternion‐valued signals using special techniques to overcome these two problems. In this context, we intend to generalize the Heisenberg uncertainty principles associated with covariance and Hardy's uncertainty principle for octonion multivector valued signals over using the polar form of an octonion, the quaternion decomposition, and the relationship between the OFT and the three‐dimensional (3D)‐Clifford‐Fourier transform.

Hardy Uncertainty Principle for the Linear Schrödinger Equation on Regular Quantum Trees

Hardy Uncertainty Principle for the Linear Schrödinger Equation on Regular Quantum Trees

Dynamical versions of Hardy’s uncertainty principle: A survey

The Hardy uncertainty principle says that no function is better localized together with its Fourier transform than the Gaussian. The textbook proof of the result, as well as one of the original proofs by Hardy, refers to the Phragmén–Lindelöf theorem. In this note we first describe the connection of the Hardy uncertainty to the Schrödinger equation, and give a new proof of Hardy’s result which is based on this connection and the Liouville theorem. The proof is related to the second proof of Hardy, which has been undeservedly forgotten. Then we survey the recent results on dynamical versions of Hardy’s theorem.

Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization

We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke–Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho–Stark principle from the point of view of quantum polarity.

Uniqueness for discrete Schrödinger evolutions

We prove that if a solution of the discrete time-dependent Schrödinger equation with bounded potential decays fast at two distinct times then the solution is trivial. For the free Schr¨odinger operator, as well as for operators with compactly supported time-independent potentials, a sharp analog of the Hardy uncertainty principle is obtained, using an argument based on the theory of entire functions. Logarithmic convexity of weighted norms is employed in the case of general bounded potentials.

Uncertainty principles for locally compact quantum groups

In this paper, we prove the Donoho–Stark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the Hirschman–Beckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections.

Gaussian decay of harmonic oscillators and related models

We prove that the decay of the eigenfunctions of harmonic oscillators, uniform electric or magnetic fields is not stable under 0-order complex perturbations, even if bounded, of these Hamiltonians, in the sense that we can produce solutions to the evolutionary Schrödinger flows associated to the Hamiltonians, with a stronger Gaussian decay at two distinct times. We then characterize, in a quantitative way, the sharpest possible Gaussian decay of solutions as a function of the oscillation frequency or the strength of the field, depending on the Hamiltonian which is considered. This is connected to the Hardy's Uncertainty Principle for free Schrödinger evolutions.

Weak Values, the Reconstruction Problem, and the Uncertainty Principle

Closely associated with the notion of weak value is the problem of reconstructing the post-selected state: this is the so-called reconstruction problem. We show that the reconstruction problem can be solved by inversion of the cross-Wigner transform, using an ancillary state. We thereafter show, using the multidimensional Hardy uncertainty principle, that maximally concentrated cross-Wigner transforms corresponds to the case where a weak measurement reduces to an ordinary von Neumann measurement.

Hardy Uncertainty Principle, Convexity and Parabolic Evolutions

We give a new proof of the $L^2$ version of Hardy's uncertainty principle based on calculus and on its dynamical version for the heat equation. The reasonings rely on new log-convexity properties and the derivation of optimal Gaussian decay bounds for solutions to the heat equation with Gaussian decay at a future time. We extend the result to heat equations with lower order variable coefficient.

Two geometric interpretations of the multidimensional Hardy uncertainty principle

Hardy's uncertainty principle says that a square integrable function and its Fourier transform cannot be simultaneously arbitrarily sharply localized. We show that a multidimensional version of this uncertainty principle can be best understood in geometrical terms using the fruitful notion of symplectic capacity, which was introduced in the mid-eighties following unexpected advances in symplectic topology (Gromov's non-squeezing theorem). In this geometric formulation, the notion of Fourier transform is replaced with that of polar duality, well-known from convex geometry. • We state a multidimensional version of Hardy's uncertainty principle. • The Hardy uncertainty principle is equivalent to a statement about the symplectic capacity of the Hardy ellipsoid. • We express this result in terms of the projections of the Hardy ellipsoid on the x - and p -spaces.

Sharp Hardy uncertainty principle and gaussian profiles of covariant Schrödinger evolutions

We prove a sharp version of the Hardy uncertainty principle for Schrödinger equations with external bounded electromagnetic potentials, based on logarithmic convexity properties of Schrödinger evolutions. We provide, in addition, an example of a real electromagnetic potential which produces the existence of solutions with critical gaussian decay, at two distinct times.

On the Prolate Spheroidal Wave Functions and Hardy’s Uncertainty Principle

We prove a weak version of Hardy's uncertainty principle using properties of the prolate spheroidal wave functions (PSWFs). We describe the eigenvalues of the sum of a time limiting operator and a band limiting operator acting on L2(R). A weak version of Hardy's uncertainty principle follows from the asymptotic behavior of the largest eigenvalue as the time limit and the band limit approach infinity. An asymptotic formula for this eigenvalue is obtained from its well-known counterpart for the prolate integral operator.

Hardy uncertainty principle and unique continuation properties of covariant Schrödinger flows

We prove a logarithmic convexity result for exponentially weighted L2-norms of solutions to electromagnetic Schrödinger equation, without needing to assume smallness of the magnetic potential. As a consequence, we can prove a unique continuation result in the style of the Hardy uncertainty principle, which generalizes the analogous theorems which have been recently proved by Escauriaza, Kenig, Ponce and Vega.

The sharp Hardy uncertainty principle for Schrödinger evolutions

We give a new proof of Hardy's uncertainty principle, up to the end-point case, which is only based on calculus. The method allows us to extend Hardy's uncertainty principle to Schr\"odinger equations with non-constant coefficients. We also deduce optimal Gaussian decay bounds for solutions to these Schr\"odinger equations.

Vector-valued distributions and Hardy’s uncertainty principle for operators

Suppose that f is a function on R n such that exp(a|·| 2 )f and exp(b|·| 2 ) ˆ f are bounded, where a,b > 0. Hardy's Uncertainty Principle as- serts that if ab > � 2 , then f = 0, while if ab = � 2 , then f = cexp( a|·| 2 ). In this paper, we generalise this uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the diagonal if the spectrum is also localised.

The Hardy uncertainty principle revisited

We give a real-variable proof of the Hardy uncertainty principle. The method is based on energy estimates for evolutions with positive viscosity, convexity properties of free waves with Gaussian decay at two different times, elliptic L 2 -estimates and the invertibility of the Fourier transform on L 2 (R n ) and S 0 (R n ).